How much is Classical Hebrew (in which the Hebrew Bible is written) is different from the modern Hebrew? In which areas lie the main differences? Can any Jew or Israeli today understand what is written in the Hebrew Bible freely or he needs a special training for that? Can anyone, please, provide here an overview?
The Classical Hebrew is more compact than modern Hebrew. The are no redundant words in Torah, hence you can see usage of "ובלכתך" instead of "כאשר אתה תלך" and alike.
A Jew (even not a religious one) can understand most parts of the Torah without special training. More problematic parts are description of offerings with much details of how and where each type of offering should be done.
Source: personal impression.
Another difference is that classical Hebrew generally uses a VSO word order (verb, subject, object), while modern Hebrew is generally SVO. As an example, the common phrase in the Torah, וידבר ה' אל משה, would literally translate as "and He spoke, G-d, to Moses"; a modern Hebrew speaker would probably say something like ה' דיבר אל משה.
for a proper overview I would probably send you to Hebrew wikipedia, but I'm guessing you (and most readers) are English speakers, so instead read: Revival Of the Hebrew Language on wikipedia.
As a side note I'll add, as a native (Israeli) Hebrew speaker that an incredibly great majority of the Hebrew Bible is readily understandable with no further sources. As other answers point out several parts are harder to understand with particular examples which pop to mind:
- Leviticus - full of "gory" details of sacrifice and sacrificial procedures
- Several scriptures which are mainly not in Hebrew or incorporate subjects which are non-Hebrew: Esther, Daniel (mostly Aramaic IIRC)
Hope that helps